Unit 1 asics of Geometry Objective- the students will be able to use undefined terms and definitions to work with points, lines and planes. Undefined Terms 1. Point has no dimension, geometrically looks like a dot, represented by a capital letter. Read point. 2. Line extends indefinitely, represented by two points or a scripted letter. Read line, Y. Y 3. Plane is a flat surface, has no thickness, represented by a capital letter. Read Plane M. M Space set of all points. ollinear points a set of points that lie on one line. oplanar points - a set of points that lie on one plane. Here are different ways of expressing relationships between points, lines, and planes. lies in k lies on k k contains k is drawn through k m and n intersect at P m and n intersect in P the intersection of m and n is P P m n
S and h lie in Z Z contains S and h S h Z Terminology Postulate (axiom) is a statement (basic assumption) assumed to be true without proof. Theorem is a statement that has to be proved. orollary is a special case of a theorem. asic ssumptions, the first line contains at least two points, a plane contains at least three points not all on one line, space contains at least 4 points not all n one plane. Through any two points there is exactly one line. Through any three points not on one line there is exactly one plane. If two points lie in a plane, then the line joining them is in that plane. If two planes intersect, then their intersection is a line. Subsets of a Line Segment - Given any two points and Y, segment Y is the set of all points consisting of and Y and all the points that lie between and Y. and Y are the endpoints. Y
Ray- Ray RS, denoted by RS is the union of RS and all the points for which it is true that S lies between R and. Opposite Rays - R S!## SR and ST are called opposite rays if S lies on RT between R and T S R T ongruent segments- segments with equal lengths. The symbol for length of a is. If = Y, then! Y Midpoint of a segment- Point M is the midpoint of if M lies on and M = M M isector of a segment- a line, segment, ray, or plane that intersects at its midpoint is a bisector of Segment ddition Postulate- If lies on, then + =
!##!!! Symbols line containing and ray with endpoint, through segment joining and length of Postulate - For any two points there is a unique positive number called the distance between the points Ruler Postulate The points on a line can be paired with the Real Numbers in such a way that: a. any desired point can be paired with zero; b. the distance between any two points is equal to the absolute value of the difference of the numbers paired with those points. Midpoint Find the coordinate of the midpoint of a segment that connects (10,0) and (12, 0). Draw a number line and find the middle of the two coordinates. 10 + 12 Using arithmetic = 22 2 2 the midpoint is (11,0) Find the coordinates of the midpoint of a segment with (2,3) and (4,5) as the endpoints. Draw a number line, find the middle of the line segment connecting 2 and 4.! 2 + 4 Using arithmetic, we have 2, 3 + 5 $ # 2 % & '! 6 2, 8 $ # 2 % & or( 3,4 )
Midpoint Formula Those examples suggest the midpoint formula-! midpoint = x + x 1 2 # 2, y 1 + y 2 2 $ % & Distance Formula d = x 2! x 1 ) 2 + (y 2! y 1 ) 2 Find the distance between (4,7) and (8,3) d = x 2! x 1 ) 2 + (y 2! y 1 ) 2 d = (8! 4) 2 + (3! 7) 2 d = 4 2 + (!4) 2 d = 16 + 16 d = 32 = 16 2 d = 4 2 Reasoning Logic Deductive reasoning Inductive reasoning- (deduce) reasoning by which a conclusion is reached based on accepted statements. conclusions are based on observing individual cases and then stating a general principle.
Objective- students will learn how to write angles, learn definitions and classification of angles and angle pairs ngle - is the union of two rays with a common endpoint. The common endpoint is called the vertex. vertex Y 3 Ways to Name an ngle 1. Name an angle by the vertex -! 1 2. Name an angle by three letters, one point on each ray and the vertex being the middle! or! 3. Name the angle by a number written in the interior of the angle!1 ngle lassification cute angles are angles less than 90º. Right angles are angles whose measure is 90º. Obtuse angles are greater than 90º, but less than 180º. Straight angles measure 180º.
For every angle there is a unique number between 0 and 180 called the measure of the angle. Protractor Postulate- The set of rays which have a common endpoint O can be paired with the numbers between 0 and 180 inclusive in such a way that; a. one of the rays is paired with zero and the other is paired with 180; b. if O is paired with x and O is paired with y, the m! O = x y ngle ddition Postulate- If lies on the interior of! O, then m! O + m! O = m! O If point lies in the interior of! O, then m! O + m! O = m! O. O The ngle ddition Postulate just indicates the sum of the parts is equal the whole. That just seems to make sense. If m!o = 30 and m!o = 15, find the m!o. m!o + m!o = m!o 30 +15 = m!o 45 = m!o N.. The parallel construction of these angle postulates and the segment postulates.
ngle Pairs djacent angles are two angles that have a common vertex, a common side, and no common interior points.! and! are adjacent angles. They have a common vertex, they have a common side and no common interior points. ngle bisector; is said to be the bisector of! if lies on the interior of! and m! = m!. bisects!. If m! = 110, find the m! m! + m! = m! ; m! = m! m! + m! = 110 2m! = 110 m! = 55 bisects!.! = 6x + 5 and! = 2x + 13, find the value of x m! = m! 6x + 5 = 2x + 13 4x = 8 x = 2
omplementary angles are two angles whose sum is 90. If! = 30, then the complement of! measures 60. Supplementary angles are two angles whose sum is 180 If m! M = 100 and m! S = 80, then! M and! S are supplementary angles. Find the value of x, if! and! are complementary! s and! = 3x and! = 2x + 10.! +! = 90 3x + (2x + 10) = 90 5x + 10 = 90 5x = 80 x =16 Theorem If the exterior sides of two adjacent angles lie in a line, they are supplementary. These angles,! 1 and! 2, are called linear pairs. 1 2 Vertical ngles The mathematical definition of vertical angles is: two angles whose sides form pairs of opposite rays. 1 2! 1 and! 2 are a pair of vertical angles.